Find the critical points of the following function. Use the Second Derivative Te

Question

Find the critical points of the following function. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum local minimum, or saddle point. Confirm your results using a graphing utility What are the critical points? Select the correct choice below and fill in any answer boxes within your choice. OA. The ortical point's) is/are 0 B. There are no critical points. dentify any local maxima. Select the correct choice below and fill in any answer boxes within your choice O A. There are local maxima at O B- There are no local maxima. Identify any local minima. Select the correct choice below and fill in any answer boxes within your choice. 0 A. There are local minima at 0 B. There are no local minima. dentify any saddle points. Select the correct choice below and fill in any answer boxes within your choice. o A. There are saddle points at 0 B. There are no saddle points. Type an ordered pair. Use a comma to separate answers as needed.) Type an ordered pair. Use a comma to separate answers as needed.) Type an ordered pair. Use a comma to separate answers as needed.) Type an ordered pair. Use a comma to separate answers as needed)

Explanation / Answer

given f(x,y)=x4+y4-32x+4y+3

fx(x,y)=4x3-32, fy(x,y)=4y3+4

for crtical points  fx(x,y)=0, fy(x,y)=0

=>4x3-32=0,4y3+4=0

=>x3=8,y3=-1

=>x=2, y=-1

the critical point is (2,-1)

fx=4x3-32, fy=4y3+4

fxx=12x2, fyy=12y2, fxy=0,D= (fxxfyy)-(fxy)2

at (2,-1)

fxx=12*22=48, fyy=12(-1)2=12, fxy=0,D=48*12 = 576

D>0, fxx>0

=>there is local minima at (2,-1)

there are no local maxima

there are no saddle points

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